Analysis, Simulation, and Applications of Stochastic Systems

随机系统是随机干扰起重要作用的系统。该项目源于网络系统、无线通信、信号处理、经济学和生态学中新兴和现有的应用，涵盖了具有不确定性、在不同配置和复杂结构之间切换的动态演化随机系统的研究。感兴趣的网络系统包括金融、通信、社会、生物和生态网络。工作将致力于学习随机网络系统的内在特性，发展数学模型和新的数学方法来分析这些系统，并设计有效的计算方案来优化和控制这些系统以达到预期的目标。研究结果对经济学、非线性系统辨识与估计、无人驾驶车辆等多智能体系统、生态系统生物多样性、社会网络等领域具有重要的应用价值。该项目将涉及本科生和研究生，并将研究与教学和学生培训相结合。这项工作将共同促进数学理论、计算方法和应用的进一步发展，以及数学教育的改进。受广泛应用的激励，本项目将研究以下研究课题。(1)开发和分析随机切换的随机模型。该系统的新特征包括(i)具有可数状态空间的依赖于过去的切换，以及（ii）具有非局部算子、有限切换集和σ有限跳变措施的切换跳扩散。将获得复发、正复发和遍历性的标准。(2)研究白噪声扰动下扩散为简并的kolmogorov型系统。将研究在环境保护区、传染病和生态控制方面的应用。(3)将开发新的切换扩散和随机逼近算法，并研究它们的收敛速度。(i)对于切换扩散解的milstein型算法，将证明该算法与它们的扩散对应物保持1阶收敛速率。（ii）受多智能体系统、共识和群体应用的激励，随机逼近算法的新颖性包括包含状态依赖的切换、状态依赖的观察噪声和一般的时变非线性函数。(4)利用量化观测得到Hammerstein非线性系统辨识的精确误差估计。将证明估计以指数小的概率从真参数的一个小邻域逃逸。(5)得到了强逼近意义下重复删除随机网络近似方案的精确误差界。这将对随机动态图的研究和社交网络的应用产生影响。将进行广泛的数值实验和模拟，以补充分析和算法设计。这些项目将涉及本科生和研究生的参与。

Stochastic systems are systems in which random disturbances play a significant role. Stemming from emerging and existing applications in networked systems, wireless communications, signal processing, economics, and ecology, this project encompasses the study of dynamically evolving stochastic systems with uncertainties, switching among different configurations, and complex structures. The networked systems of interest include financial, communication, social, biological, and ecological networks. The work will be devoted to learning the intrinsic properties of stochastic network systems, developing mathematical models and novel mathematical methods for analyzing such systems, and designing efficient computational schemes for optimization and control of such systems to meet desired goals. The results of the research will be useful for applications to economics, nonlinear system identification and estimation, un-manned vehicles and other multi-agent systems, biodiversity in ecological systems, and social networks. This projects will involve undergraduate and graduate students and will integrate the research with teaching and student training. This work will contribute jointly to the further development of mathematical theory, computational methods and applications, and the improvement of mathematics education. Motivated by a wide variety of applications, this project will study the following research topics. (1)Stochastic models with random switching will be developed and analyzed. Novel features of the systems include (i) past-dependent switching having a countable state space, and (ii) switching jump diffusions with non-local operators, finite switching set, and sigma finite jump measures. Criteria for recurrence, positive recurrence, and ergodicity will be obtained. (2) Kolmogorov-type systems under white noise perturbations, where the diffusions are degenerate, will be investigated. Applications to control dependent environmental protection zones, infectious disease and ecology will be studied. (3) New algorithms for switching diffusions and stochastic approximation will be developed and their rates of convergence will be studied. (i) For Milstein-type algorithms for solutions of switching diffusions, it will be shown that the algorithms preserve order 1 convergence rates as their diffusion counterpart. (ii) Motivated by applications to multi-agent systems, consensus, and swarming, the novelties of the stochastic approximation algorithms include the inclusion of state-dependent switching, state-dependent observation noise, and general time-dependent nonlinear functions. (4) Precise error estimates for identification of Hammerstein nonlinear systems with quantized observations will be obtained. It will be proved that the estimates escape from a small neighborhood of the true parameter with a probability that is exponentially small. (5) Accurate error bounds for approximation schemes of duplication-deletion random networks in the sense of strong approximation will be obtained. This will have impact on the study of random dynamic graphs and applications to social networks. Extensive numerical experiments and simulations will be performed to complement the analysis and algorithm design. The projects will involve the participation of undergraduate and graduate students.